1 Introduction
Coalition formation plays a major role in a broad range of settings. Whether politicians forming political parties to participate in an election, countries joining alliances to increase their negotiation power, or students getting together in groups for a classroom project, one can observe coalition formation at work. It is therefore important to understand the process of how coalitions form: when can we expect coalitions to form, and when are the coalition members “happy” with their coalitions? Within game theory, the importance of coalitions became clear since von Neumann and Morgenstern’s seminal work
Theory of Games and Economic Behavior was published in 1944.The hedonic viewpoint of coalition formation was introduced by Drèze and Greenberg (1980), who first called attention to the “hedonic aspect” of the game, i.e., the determination of a player’s utility by the coalition that the player belongs to. Banerjee et al. (2001) and Bogomolnaia and Jackson (2002) introduced and analyzed stability concepts in hedonic coalition formation games, including Nash stability, individual stability, and core stability. Among other things, Bogomolnaia and Jackson defined restricted classes of hedonic games called additively separable hedonic games and symmetric additively separable hedonic games, and showed that a Nash stable partition always exists in the latter class of games but not necessarily in the former. Since then, several restrictions on hedonic games have been proposed, including hedonic games based on best players and hedonic games based on worst players by Cechlárová and RomeroMedina (2001) and fractional hedonic games by Aziz et al. (2014). Other authors who have proposed and analyzed restrictions on hedonic games include Alcalde and Revilla (2004), Alcalde and RomeroMedina (2006), Burani and Zwicker (2003), and Dimitrov et al. (2006). For an excellent survey on hedonic games, we refer to Hajduková (2006).
In this paper, we define the class of subsetadditive hedonic games, which generalizes the class of additively separable hedonic games. We show that our class does not provide any restriction on the game – any hedonic game is also a subsetadditive hedonic game. We then define the class of subsetneutral hedonic games, which is a restriction of subsetadditive hedonic games and a generalization of symmetric additively separable hedonic games. Even though subsetneutral hedonic games enjoy significantly more representation power than symmetric additively separable hedonic games, they also provide a guarantee of the existence of a Nash stable partition and an individually stable partition. We also consider neutrally anonymous hedonic games, which is a somewhat restricted class of games but can still model many interesting situations. We show that they form a subclass of the subsetneutral hedonic games, hence inheriting the guarantee of the existence of a Nash stable partition and an individually stable partition. Finally, we show that a core stable partition that is also individually stable is guaranteed to exist in neutrally anonymous hedonic games by exhibiting an algorithm that computes such a partition.
2 Definitions and notation
In this section, we introduce the setting and give definitions and notation that we will use throughout this paper.
Let be a finite set of players in the game. Denote by the set of all subsets of that include . A coalition is a nonempty subset of . A coalition partition is a partition of the set into disjoint coalitions. Denote by the coalition in that belongs to. For any set , denote by the set of all subsets of and the size of .
We assume throughout the paper that the players’ preferences are hedonic, i.e., they are completely determined by the coalition that the player belongs to. Each player is endowed with a preference relation , a reflexive, complete, and transitive ordering over . Let denote the strict part and the indifference part of the relation . A hedonic coalition formation game is represented by a pair .
We now define stability notions that we will consider in the paper. The following three stability notions, which consider deviations by a single player, were introduced by Bogomolnaia and Jackson (2002).
Definition 1
A coalition partition is Nash stable if for all and .
Definition 2
A coalition partition is individually stable if for any and such that , there exists a player such that .
Definition 3
A coalition partition is contractually individually stable if for any and such that , there exists either a player such that , or a player such that and .
Any Nash stable partition is also individually stable and any individually stable partition is also contractually individually stable. Ballester (2004) showed that every hedonic game contains at least one contractually individually stable partition.
Next, we define two stability notions that consider deviations by a coalition.
Definition 4
A coalition blocks a coalition partition if for all . A coalition partition is core stable if any coalition does not block .
Definition 5
A coalition weakly blocks a coalition partition if for all and for some . A coalition partition is strong core stable if any coalition does not weakly block .
Any strong core stable partition is also core stable as well as individually stable, and neither Nash stability nor core stability implies the other.
We now define properties on preference profiles.
Definition 6
A preference profile is separable if

if and only if , and

if and only if
for all such that and .
Definition 7
A preference profile is additively separable if for all , there exists a function such that
for all .
Furthermore, an additively separable preference profile is called symmetric if for all , and is called mutual if for all , whenever .
Definition 8
A preference profile is anonymous if for any and any two coalitions such that and .
Definition 9
A preference profile satisfies the common ranking property if there exists a function such that for all ,
for all .
Definition 10
Given a nonempty set , a nonempty subset is a top coalition of if for any and any with , we have . A preference profile satisfies the topcoalition property if for any nonempty set , there exists a top coalition of .
3 Subsetadditive and subsetneutral hedonic games
In this section, we define a generalization of additively separable games that we call subsetadditive hedonic games, and show that the class of subsetadditive hedonic games in fact coincides with the class of hedonic games. We then define a generalization of symmetric additively separable games that we call subsetneutral hedonic games. We show that subsetneutral hedonic games have more representation power than symmetric additively separable games, and we generalize a result by Bogomolnaia and Jackson (2002) by proving that the existence of a Nash stable partition and an individually stable partition is guaranteed in subsetneutral hedonic games.
We start with the definition of subsetadditive preference profiles.
Definition 11
A preference profile is subsetadditive if for all , there exists a function such that
It turns out that subsetadditivity does not provide a restriction on the preference profile, as the following proposition shows.
Proposition 1
Any hedonic game is also a subsetadditive hedonic game.
Proof
Consider a preference profile of a hedonic game. For all , let be a utility function consistent with player ’s preference profile. We define the function recursively from smaller to larger sets in . If has been defined for all , we define
It follows that
which implies that the game is subsetadditive, as desired.
Although subsetadditivity provides no restriction on the preference profile, if we impose a neutrality condition on the utility function, we obtain a smaller class of preference profiles.
Definition 12
A preference profile is subsetneutral if there exists a function such that
for all .
Subsetneutral preference profiles are useful for modeling situations in which different teams can form within a coalition. For example, suppose that a police department is divided into different subdivisions, which correspond to our coalitions. Certain teams of police officers will be assigned by the chief to tackle a criminal case based on their combined specialty if they belong to the same subdivision. The value of a subdivision to a police officer is therefore the sum of the values of the different teams to which he will be assigned to work on cases. Since team chemistry varies according to the composition of the team, one can imagine that the value is different for different teams and cannot be broken down into values between pairs as in additively separable preference profiles. Subsetneutral preference profiles also allow for situations in which the chief assigns as many or as few teams as he likes, since teams that are not assigned simply correspond to a value of 0.
If we set for all in Definition 12, we obtain the class of symmetric additively separable preference profiles. Hence any symmetric additively separable preference profile is also subsetneutral. On the other hand, not all subsetneutral profiles are additively separable (or even separable), as the following example shows.
Example 1
Consider the game with and the function given by

for all ;

;

.
We have and , which violates separability.
A preference profile satisfying subset neutrality cannot have cycles on coalitions of size 2 between different players. For instance, it cannot be the case that , and hold simultaneously. On the other hand, cycles on coalitions of size greater than 2 between different players can occur, as the following example shows.
Example 2
Consider the game with and the function given by

for all ;

;

.

The value of on other subsets can be defined arbitrarily.
We have and .
Bogomolnaia and Jackson (2002) showed that for symmetric additively separable hedonic games, a Nash stable partition and an individually stable partition exist. The next theorem generalizes that result.
Theorem 3.1
For subsetneutral hedonic games, a Nash stable partition and an individually stable partition exist.
Proof
Since a Nash stable partition is also individually stable, it suffices to show that a Nash stable partition exists.
Consider a partition that maximizes the potential function
such a partition exists since the total number of partitions is finite. We claim that is a Nash stable partition. Suppose for contradiction that player has an incentive to move from coalition to a different coalition . This means that the utility that gains from the subsets in is greater than the utility that gains from the subsets in , i.e.,
Let be the partition that is obtained from if moves from coalition to . The potential function of the partition is
contradicting the assumption that is a partition that maximizes the potential function.
The proof of Theorem 3.1 relies crucially on neutrality. Bogomolnaia and Jackson (2002) showed that an individual stable partition (and hence a Nash stable partition) may not exist even if preference profiles are additively separable, mutual, and single peaked on a tree. (For the definition of singlepeakedness on a tree, we refer to their paper.) On the other hand, symmetry in additively separable hedonic games is not enough to guarantee the existence of a core stable partition. Indeed, Banerjee et al. (2001) showed that a core stable partition (and hence a strong core stable partition) may not exist even if preference profiles are additively separable and symmetric.
4 Neutral anonymity
In this section, we define a restriction of anonymous hedonic games that we call neutrally anonymous hedonic games. We show that such games are subsetneutral, and hence existence of a Nash stable partition and an individually stable partition is guaranteed by results in Section 3. We then show that a partition that is both core stable and individually stable must exist in neutrally anonymous hedonic games, and in fact, in the more general class of games whose preference profiles satisfy the common ranking property. We exhibit an algorithm to find such a partition.
We start with the definition of neutrally anonymous preference profiles.
Definition 13
A preference profile is neutrally anonymous if there exists a function such that for all ,
Note that the function in the definition takes on the cardinality of the set of players rather than the set of players itself.
Any neutrally anonymous preference profile is also anonymous and satisfies the common ranking property. On the other hand, a neutrally anonymous preference profile need not be separable or singlepeaked on the size of the coalition to which the player belongs. Indeed, consider the neutrally anonymous hedonic game with and the function given by , and . This preference profile is neither separable nor singlepeaked on the size of the coalition to which the player belongs.
Even though neutral anonymity is somewhat restrictive, there are interesting situations that can be modeled using preference profiles satisfying this property. For instance, a teacher may assign a different amount of work in a classroom project to groups of students of different sizes. If the students are only concerned about the amount of work that they have to do, their utility will only depend on the size of their group. As another example, a restaurant may provide a different amount of food or issue a different price depending on the size of the party. If the utility of each customer is determined by the foodtoprice ratio, the situation can again be modeled with a neutrally anonymous hedonic game. One can imagine that in such situations, the preference profile need not be monotonic or even singlepeaked in the size of the coalition to which the player belongs.
It turns out that any neutrally anonymous hedonic game is also a subsetneutral hedonic game, as the following proposition shows.
Proposition 2
Any neutrally anonymous preference profile is also subsetneutral.
Proof
Consider an arbitrary neutrally anonymous game, and let be the function associated to it as in Definition 13. We will exhibit a function associated to it as in Definition 12 such that
for all .
We define the function recursively from smaller to larger subsets of . Because of neutral anonymity, sets of the same size have the same value of the function . Hence it suffices to define the function for the sets . If has been defined for all such that , we define
It follows that
for all , as desired.
Theorem 4.1
For neutrally anonymous hedonic games, a Nash stable partition and an individually stable partition exist.
In contrast to Theorem 4.1, Bogomolnaia and Jackson (2002) showed that a Nash stable partition may not exist even if the preference profile is anonymous and singlepeaked. The next example shows that a Nash stable partition may not exist even if the preference profile satisfies the common ranking property.
Example 3
Consider the game with and the common ranking given by
Let be any partition of . If player 3 is left alone in , she will have an incentive to join one of the existing coalitions. Otherwise, player 3 is in the same coalition in as at least one of player 1 and player 2, and that player has an incentive to form a coalition by herself.
The next theorem shows that a core stable partition that is also individually stable and contractually individually stable exists in games whose preference profiles satisfy the common ranking property. On the other hand, a strict core stable partition may not exist even in neutrally anonymous hedonic games, as shown by the game with and the function given by , and .
Theorem 4.2
For games whose preference profiles satisfy the common ranking property, a core stable partition that is also individually stable and contractually individually stable exists. Moreover, we exhibit an algorithm to compute such a partition.
Proof
Since individual stability implies contractually individual stability, it suffices to give an algorithm to compute a core stable partition that is also individually stable. The algorithm operates as follows:

Initially, let be the set of all players.

Repeat the following until :

Choose a coalition that ranks highest in the common ranking of all the nonempty subsets of . If there are several such coalitions, choose one with the largest size.

Remove the players in from .

The algorithm clearly terminates. We first show that the resulting partition is core stable. We prove by induction that when a coalition is formed by the algorithm, every player in the coalition is unwilling to participate in a blocking coalition. This is true for the first coalition formed, since the players in the coalition rank the coalition highest in their ranking. For any subsequent coalition formed, a player in the coalition rank the coalition highest among the coalitions that she can form using her coalition and the remaining players. By the induction hypothesis, no player in previous coalitions is willing to participate in a blocking coalition. Hence the players in the coalition formed also do not want to participate in a blocking coalition, completing the induction.
We now show that the resulting partition is individually stable. Consider a deviation by player . Again, she is already in a coalition that ranks highest among the coalitions that she can form using her coalition and the remaining players. Hence she has no incentive to switch to a coalition formed after her. If she switches to a coalition formed before her, then since that coalition is a largest coalition that ranks highest when it is formed, the inclusion of player necessarily leaves the members of that coalition worse off. Finally, player has no incentive to form a coalition on her own if she is not already in a coalition by herself. Hence the partition is individually stable, as desired.
The algorithm in Theorem 4.2 is a specific version of the algorithm proposed by Banerjee et al. (2001) for finding a core stable partition when the preference satisfies the topcoalition property. While our algorithm requires a stronger condition, it produces a partition that is both core stable and individually stable, instead of only core stable. The crucial difference between the two algorithms is that our algorithm requires in Step 2 that if there are several coalitions that rank highest among the remaining coalitions, our algorithm chooses a coalition with the largest size. If we eliminate this requirement, the resulting partition is no longer guaranteed to be individually stable, as the following example shows.
Example 4
Consider the game with and the common ranking given by
Initially, and are the coalitions that rank highest in the common ranking. If the algorithm chooses the coalition as the first coalition, the resulting partition will be . This partition is not individually stable, however, since player 3 can benefit by joining the other two players while not leaving them worse off.
In addition, if we assume only the topcoalition property, then Banerjee et al.’s algorithm (or any specific version of it) is not guaranteed to return an individually stable partition, as the following example shows.
Example 5
Consider the game with and the preference profile given by

;

;

.
The preference profile satisfies the topcoalition property. Indeed, is a top coalition of for all , is a top coalition of and , is a top coalition of , and is a top coalition of .
Since is the unique top coalition in , Banerjee et al.’s algorithm (or any specific version of it) will choose in the first step, yielding the partition . This partition is not individually stable, however, since player 3 can benefit by joining the other two players while not leaving them worse off.
We obtain the following as a corollary of Theorem 4.2.
Corollary 1
For neutrally anonymous hedonic games, a core stable partition that is also individually stable and contractually individually stable exists.
In contrast to Theorem 4.2, Banerjee et al. (2001) showed that a core stable partition may not exist even if the preference profile is anonymous, singledpeaked on population, and satisfies the population’s intermediate preference property. (For definitions, we refer to their paper.)
Although a core stable partition produced by the algorithm in Theorem 4.2 is guaranteed to be individually stable, this is not necessarily the case for any core stable partition in neutrally anonymous hedonic games. In fact, a core stable partition need not even be contractually individually stable in such games, as the following example shows.
Example 6
Consider the game with and the function given by and .
The partition is core stable but not contractually individually stable, since player 3 can join the coalition without leaving any member of her old or new coalition worse off. In this example, the grand coalition is both core stable and individually stable.
Theorems 4.1 and 4.2 guarantee the existence of both a Nash stable partition and a core stable partition in neutrally anonymous hedonic games. Nevertheless, there may not exist a partition that is both Nash stable and core stable, as the following example shows.
Example 7
Consider the game with and the function given by , , , and .
Any core stable partition cannot contain a coalition of size 3, 4, or 5, since two players from such a coalition form a blocking coalition. A partition that consists only of coalitions of size 1 and 2, however, necessarily contains a player in a coalition by herself. Such a player has an incentive to switch to any existing coalition, which implies that such a partition cannot be Nash stable.
Example 7 also shows that neither Nash stability nor core stability implies the other even in neutrally anonymous hedonic games.
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